A remark is in order concerning the case \(a=0\) , which corresponds of having only gravitons in the theory, as we see from \(S_{inv}(a)\)  (REF ). For \(a=0\)  we have only the constraint () on the symmetric tensor field \(A_{\mu \nu }(x)\) , which therefore would display 9 dof. Now, we know that the dof of massless LG are, instead, 6 [1]}, [2]}. It is interesting and instructive to go back to the origin of this mismatch. We already noticed that LG is a particular solution of the fracton symmetry (REF ), which is protected by the stronger infinitesimal diffeomorphism symmetry (REF ). In other words, for \(a\rightarrow 0\)  the symmetry switches from the weaker (REF ) to the stronger (REF ). This transition is not continuous as far as the gauge parameter is concerned, because it changes from being a scalar (for the fracton symmetry (REF )) to become a vector field (for the infinitesimal diffeomorphism (REF )). As a consequence, the “scalar” gauge fixing realised in \(S_{gf}\)  (REF ) is not a proper gauge fixing for the \(a=0\)  case, and one should choose instead a “vectorial” gauge fixing, as done in [3]}, [4]} for LG. This is the physical explanation of the divergence of the propagator (REF ) for \(a\rightarrow 0\) .
In the case \(a\ne 0\) , let us define
\(\tilde{J}_\rho \equiv p^\sigma \tilde{A}_{\rho \sigma }\ ,\) 
